Voozh

A091824

Numbers p such that ((p-1)!*2^(p-1) + 1)/p is a prime.

1, 3, 37, 61, 10651

OFFSET

1,2

COMMENTS

1 is not prime, but (1-1)!*2^(1-1) + 1 = 2 is a prime, so 1 is in the sequence.

If p is a prime and gcd(q,p)=1, then p divides (p-1)!*q^(p-1) + 1.

LINKS

EXAMPLE

((1-1)!*2^(1-1) + 1)/1 = 2 is a prime, so 1 is the first term in the sequence.

((2-1)!*2^(2-1) + 1)/2 = is not an integer, so 2 is not in the sequence.

((3-1)!*2^(3-1) + 1)/3 = 3 is a prime, so 3 is the second term in the sequence.

The next number p that yields a prime is 37:

((37-1)!*2^(37-1) +1)/37 = 690896939629347629014331483828706966091078572972973.

PROG

(PARI) for (i=3, 1100, if(isprime(((i-1)!*2^(i-1)+1)/i), print(i)));

(Magma) [1] cat [p: p in PrimesUpTo(10000) | IsPrime((Factorial(p-1)*2^(p-1) + 1) div p )]; // Vincenzo Librandi, Aug 21 2015

CROSSREFS

KEYWORD

hard,more,nonn

AUTHOR

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 09 2004

EXTENSIONS

a(5) from Michael S. Branicky, Apr 24 2023

STATUS

approved

URL: https://oeis.org/A091824